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Width of L^ P Balls Widths of l p balls Antoine Gournay November 2, 2007 1 Introduction Let (X,d) be a metric space and ε ∈ R>0, then we say a map f : X → Y is an ε- embedding if it is continuous and the diameter of the fibres is less than ε, i.e. ∀y ∈ Y,Diam f −1(y) ≤ ε. We will use the notation f : X ֒→ε Y. This type of maps, which can be traced at least to the work of Pontryagin (see [13] or [8]), is related to the notion of Urysohn width (sometimes referred to as Alexandrov width), an(X), see [3]. It is the smallest real number such that there exists an ε-embedding from X to a n-dimensional polyhedron. Surprisingly few estimations of these numbers can be found, and one of the aims of this paper is to present some. However, following [7], we shall introduce: Definition 1.1: wdimεX is the smallest integer k such that there exists an ε-embedding f : X → K where K is a k-dimensional polyhedron. wdimε(X,d)= inf dimK. X ֒→ε K Thus, it is equivalent to be given all the Urysohn’s widths or the whole data of wdimεX as a function of ε. Definition 1.2: The wdim spectrum of a metric space (X,d), denoted wspecX ⊂ Z≥0 ∪ {+∞}, is the set of values taken by the map ε 7→ wdimεX. The an(X) obviously form an non-increasing sequence, and the points of wspecX are precisely the integers for which it decreases. We shall be interested in the widths p l (n) Rn of the following metric spaces: let B1 be the set given by the unit ball in for the p p ∑ p 1/p l (n) ∞ l metric (k(xi)kl p = |xi| ), but look at B1 with the l metric (i.e. the sup metric of the product). Then l p(n) ∞ ε R Proposition 1.3: wspec(B1 ,l )= {0,1,...,n}, and, ∀ ∈ >0, 0 if 2 ≤ ε p l (n) ∞ −1/p ε −1/p wdimε(B1 ,l )= k if 2(k + 1) ≤ < 2k . n if ε < 2n−1/p ε l p(n) ∞ The important outcome of this theorem is that for fixed , the wdimε(B1 ,l ) is bounded from below by min(n,m(p,ε)) and from above by min(n,M(p,ε)), where ε hal-00189043, version 2 - 15 Feb 2008 m,M are independent of n. As an upshot high values can only be reached for small independantly of n. It can be used to show that the mean dimension of the unit ball of 1 l p(Γ), for Γ a countable group, with the natural action of Γ and the weak-∗ topology is zero when p < ∞ (see [14]). It is one of the possible ways of proving the non-existence of action preserving homeomorphisms between l∞(Γ) and l p(Γ); a simpler argument would be to notice that with the weak-∗ topology, Γ sends all points of l p(Γ) to 0 while l∞(Γ) has many periodic orbits. The behaviour is quite different when balls are looked upon with their natural met- ric. Theorem 1.4: Let p ∈ [1,∞), n > 1, then ∃hn ∈ Z satisfying hn = n/2 for n even, n+1 n−1 h3 = 2 and hn = 2 or 2 otherwise, such that p n {0,h(n),n}∪⊂ wspec(Bl (n),l p) ⊂{0} ∪ ( − 1,n] ∩ Z. 1 2 When p = 2 or when p = 1 and there is a Hadamard matrix of rank n + 1, then n − 1 p l (n) p also belongs to wspec(B1 ,l ). N n More precisely, let k,n ∈ with 2 − 1 < k < n. Then there exists bn;p ∈ [1,2] and ck,n;p ∈ [1,2) such that p ε l (n) p if ≥ 2 then wdimε(B1 ,l )= 0 p ε l (n) p n if < 2 then wdimε(B1 ,l ) > − 1 p 2 ε l (n) p if ≥ ck,n;p then wdimε(B1 ,l ) ≤ k p ε l (n) p if < bk;p then wdimε(B1 ,l ) ≥ k and, for fixed n and p, the sequence ck,n;p is non-increasing. Furthermore, bk;p ≥ ′ 1/p′ 1 1/p 1/p 1 1/p ∞ 2 1 + k when 1 ≤ p ≤ 2, whereas bk;p ≥ 2 1 + k if 2 ≤ p < . 1 Additionally, in the Euclidean case (p = 2), we have that bn;2 = cn−1,n;2 = 2(1 + n ), 1/p 1/p′ q while in the 2-dimensional case b2;p ≥ max(2 ,2 ) for any p ∈ [1,∞]. Also, if p = 1 1, and there is a Hadamard matrix in dimension n + 1, then bn;1 = cn−1,n;1 = 1 + n . p ε l (n) 2 1/p Finally, when n = 3, ∀ > 0,wdimεB1 6= 1 and c2,3;p ≤ 2( 3 ) , which means in particular that c2,3;p = b3;p when p ∈ [1,2]. Various techniques are involved to achieve this result; they will be exposed in sec- tion 3. While upper bounds on wdimεX are obtained by writing down explicit maps to a space of the proper dimension (these constructions use Hadamard matrices), lower bounds are found as consequences of the Borsuk-Ulam theorem, the filling radius of spheres, and lower bounds for the diameter of sets of n + 1 points not contained in an open hemisphere (obtained by methods very close to those of [9]). We are also able to give a complete description in dimension 3 for 1 ≤ p ≤ 2. 2 Properties of wdimε Here are a few well established results; they can be found in [1], [2], [11], and [12]. Proposition 2.1: Let (X,d) and (X ′,d′) be two metric spaces. wdimε has the following properties: a. If X admits a triangulation, wdimε(X,d) ≤ dimX. 2 b. The function ε 7→ wdimεX is non-increasing. c. Let X be the connectedcomponentsof X, then wdimε X d 0 ε maxDiamX . i ( , )= ⇔ ≥ i i ′ ′ ′ d. If f : (X,d) → (X ,d ) is a continuousfunction such that d(x1,x2) ≤Cd ( f (x1), f (x2)) ∞ ′ ′ where C ∈]0, [, then wdimε(X,d) ≤ wdimε/C(X ,d ). e. Dilations behave as expected, i.e. let f : (X,d) → (X ′,d′) be an homeomorphism ′ such that d(x1,x2)= Cd ( f (x1), f (x2)); this equality passes throughto the wdim: ′ ′ wdimε(X,d)= wdimε/C(X ,d ). f. If X is compact, then ∀ε > 0,wdimε(X,d) < ∞. Proof. They are brought forth by the following remarks: a. IfdimX = ∞, the statement is trivial. For X a finite-dimensional space, it suffices to look at the identity map from X to its triangulation T(X), which is continuous and injective, thus an ε-embedding ∀ε. b. If ε ≤ ε′, an ε-embedding is also an ε′-embedding. ε ∋ c. If wdimεX = 0 then ∃φ : X ֒→ K where K is a totally discontinuous space. ∀k K,φ−1(k) is both open and closed, which implies that it contains at least one connected component, consequently DiamXi ≤ ε. On the other hand, if ε ≥ DiamXi the map that sends every Xi to a point is an ε-embedding. ′ ε φ ′ d. If wdimε/CX = n, there exists an C -embedding : X → K with dimK = n. Noticing that the map φ ◦ f is an ε-embedding from X to K allows us to sustain the claimed inequality. e. This statement is a simple application of the preceding for f and f −1. f. To show that wdimε is finite, we will use the nerve of a covering; see [8, §V.9] for example. Given a covering of X by balls of radius less than ε/2, there exists, by compactness, a finite subcovering. Thus, sending X to the nerve of this finite covering is an ε-immersion in a finite dimensional polyhedron. Another property worth noticing is that lim wdimε X d dimX for compact X; ε→0 ( , )= we refer the reader to [1, prop 4.5.1]. Reading [6, app.1] leads to believe that there is a strong relation between wdim and the quantities defined therein (Radk and Diamk); the existence of a relation between wdim and the filling radius becomes a natural idea, implicit in [7, 1.1B]. We shall make a small parenthesis to remind the reader of the definition of this concept, it is advised to look in [6, §1] for a detailed discussion. Let (X,d) be a compact metric space of dimension n,and let L∞(X) be the (Banach) space of real-valued bounded functions on X, with the norm k f kL∞ = sup | f (x)|. The x∈X metric on X yields an isometric embedding of X in L∞(X), known as the Kuratowski embedding: ∞ IX : X → L (X) ′ ′ x 7→ fx(x )= d(x,x ). 3 The triangle inequality ensures that this is an isometry: sup ′′ ′ ′′ ′ k fx − fx′ kL∞ = d(x,x ) − d(x ,x ) = d(x,x ). x′′ X ∈ ∞ Denote by Uε(X) the neighborhood of X ⊂ L (X) given by all points at distance less than ε from X, ∞ i.e. Uε X f L X inf f f ∞ ε ( )= ∈ ( ) x∈X k − xkL < . Definition 2.2: The filling radius of a n-dimensional compact metric space X, written FilRadX, is defined as the smallest ε such that X bounds in Uε(X), i.e. IX (X) ⊂ Uε(X) induces a trivial homomorphism in simplicial homology Hn(X) → Hn(Uε(X)). Though FilRad can be defined for an arbitrary embedding, we will only be con- cerned with the Kuratowski embedding.
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